Small NFAs from Regular Expressions: Some Experimental Results

Regular expressions (res), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA). We present the implementation of several algorithms for constructing small epsilon-free NFAss from res within the FAdo system, and a comparison of regular expression measures and NFA sizes based on experimental results obtained from uniform random generated res. For this analysis, nonredundant res and reduced res in star normal form were considered.Comment: Proceedings of 6th Conference on Computability in Europe (CIE 2010), pages 194-203, Ponta Delgada, Azores, Portugal, June/July 201

Real Hypercomputation and Continuity

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of "Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS vol.352

Revising Type-2 Computation and Degrees of Discontinuity

By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200